So it’s no general problem and BW passkeys - in general - do work with Arc browser and the BW browser extension.
I don’t use discord, so I have no idea of their passkey implementation. – It could be their implementation… it could also be some kind of “incompatibility” between their implementation and Bitwarden…
PS: Well, wait…
Did you check on the browser extension on Arc browser, if
- “Ask to save and use passkeys” is turned on (Settings → Notifications) and
- “Discord” is not on your “Excluded domains” list (Settings → Notifications → Excluded domains) ?
PPS: And, you are on MacOS… as some reported, (my words now) MacOS and it’s passwords.app seem to be very “intrusive” and may also contribute to your issue… (see e.g. here and the following posts)